8. Determine the number M and the interval width h so that the composite trapezoidal rule for M subintervals can be used to compute the given integral with an accuracy of 5 x 10-9. r/6 1 (a) cos(x) dx (b) (c) xe dx 5- Hint for part (c). f(2) (x) = (x - 2)e-x.
8. Determine the number M and the interval width h so that the composite trapezoidal rule for M subintervals can be used to compute the given integral with an accuracy of 5 x 10-9. r/6 1 (a) cos(x) dx (b) (c) xe dx 5- Hint for part (c). f(2) (x) = (x - 2)e-x.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![7. Derive Simpson's rule (M = 1, h = 1) by using the method of undetermined coeffi-
cients.
(a) Find the constants wo, w₁, and we so that f g(t) dt = wog(0) + wig(1) +
w2g (2) is exact for the three functions g(t) = 1, g(t)=t, and g(t) = 1².
(b) Use the relation f(xo +ht) = g(t) and the change of variable x = xo +ht and
dx = hdi to translate the trapezoidal rule over [0, 2] to the interval [x0, x2].
Hint for part (a). You will get a linear system involving the three unknowns wo, wi,
and 22.
8. Determine the number M and the interval width h so that the composite trapezoidal
rule for M subintervals can be used to compute the given integral with an accuracy of
5 × 10-⁹.
<-1
(a) [16
cos(x) dx
(b) h 51²74x
dx
(c)
6²³
xe di
-π/6
Hint for part (c). f(2)(x) = (x - 2)e-*.
9. Determine the number M and the interval width h so that the composite Simpson rule
for 2M subintervals can be used to compute the given integral with an accuracy of
5 × 10-⁹.
2/6
(a)
cos(x) dx
(b)
f
dx
(c)
sỉ xe
xe đã
5-
-π/6
Hint for part (c). f(4)(x) = (x-4)e-*.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9663f5ac-8453-4c53-9401-464bec0132fb%2F20d3f6e1-bba0-4f32-94ae-f84ccbb2f5f9%2F598iu75_processed.jpeg&w=3840&q=75)
Transcribed Image Text:7. Derive Simpson's rule (M = 1, h = 1) by using the method of undetermined coeffi-
cients.
(a) Find the constants wo, w₁, and we so that f g(t) dt = wog(0) + wig(1) +
w2g (2) is exact for the three functions g(t) = 1, g(t)=t, and g(t) = 1².
(b) Use the relation f(xo +ht) = g(t) and the change of variable x = xo +ht and
dx = hdi to translate the trapezoidal rule over [0, 2] to the interval [x0, x2].
Hint for part (a). You will get a linear system involving the three unknowns wo, wi,
and 22.
8. Determine the number M and the interval width h so that the composite trapezoidal
rule for M subintervals can be used to compute the given integral with an accuracy of
5 × 10-⁹.
<-1
(a) [16
cos(x) dx
(b) h 51²74x
dx
(c)
6²³
xe di
-π/6
Hint for part (c). f(2)(x) = (x - 2)e-*.
9. Determine the number M and the interval width h so that the composite Simpson rule
for 2M subintervals can be used to compute the given integral with an accuracy of
5 × 10-⁹.
2/6
(a)
cos(x) dx
(b)
f
dx
(c)
sỉ xe
xe đã
5-
-π/6
Hint for part (c). f(4)(x) = (x-4)e-*.
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